Speeding Ticket- Should I argue?

Today I was pulled over and given a $285 speeding ticket for going 27 over the speed limit. I was speeding and I admitted it to the officer that I was speeding, but at the time I was so freaked out by pulling over that I didn’t even try to deny it. I’m not looking to get out of the punishment, really, I just want to avoid it showing up on insurance. I was recently in a car accident and I don’t even want to think about what a ticket would do to my premium.

In hindsight, I think the cop was a tad unforgiving- He was sitting in an area where it goes from 55 mph rural to 30 mph residential, and I’m a 17 year old girl who has a clean record. And yes, I was crying and freaking out and I didn’t challenge him at all. Maybe it was a really bad day, I don’t know.

As of now, my parents don’t know about this. I have the means to pay the fine and intend to, but my dad pays for my insurance so I’d really like for this not to show up, or for the effects of it to be reduced when it comes to insurance. I live in MN, where I understand that attending a Traffic School will not get tickets dismissed.

I have been researching contesting tickets in court and just don’t know what to do- I was stupid, yes, and I’m willing to pay for that, I just don’t want to be paying for it for the next 5 years and have my dad paying for it. But I know I was speeding and I admitted it to the officer, meaning that really I have no right to a not-guilty plea. But I would like to negotiate something, if possible, where I pay a fine and attend a traffic class rather than the speeding showing up on my record. Is there any way to do this outside of an official court hearing? Should I try to negotiate or should I just take it? What do you think? I just don’t know what to do- I plan on telling my father, but I really want to have a good idea of my options before doing so, so we can skip past the “I messed up” part of the conversation to the ”....But I’m trying to be responsible for it and here’s how I plan to do that.” part.
I’m 17, have an otherwise clean record, and maintain a B-A average in school.

Also, our insurance expired on the 10th- How long until the insurance company can see this ticket?